Computer Algebra Recipes

6.1. WAVE EQUATION MODELS 249
> sol:=pdsolve(WE,HINT=sin(k*x)*g(t),INTEGRATE,build);
sol := Ã(x; t) =sin(kx) C1 sin(ckt)+sin(kx) C2 cos(ckt)
The string is initially at rest, so the transverse velocity _ Ã(x; 0) is equal to 0.
Mentally di®erentiating the above output, clearly only the cos(ckt) term should
be retained, since its derivative vanishes at t = 0, whereas the derivative of the
sin(ckt) term does not. So Vectoria selects the term on the rhs of the solution
containing the cosine term.
> sol2:=select(has,rhs(sol),cos);
sol2 := sin(kx) C2 cos(ckt)
The displacement of the string must also vanish at x = L for arbitrary times, so
onemusthavesin(kL) = 0, which yields k = n¼=L,withnapositive integer.
She substitutes this result into sol2 , and temporarily removes any coe±cients
by setting C1 and C2 equal to 1.
> F[n]:=subs(fk=n*Pi/L,_C1=1,_C2=1g,sol2);
³
n¼x
´ μ
cn¼t
Fn := sin cos
L
L
The general solution satisfying the boundary conditions at x =0andL, and
with zero initial transverse velocity, then is
1X
1X ³
n¼x
´ μ
cn¼t
Ã(x; t) = An Fn = An sin cos ; (6.2)
L
L
n=1
n=1
where the coe±cients An have to be determined from the initial string pro¯le.
This pro¯le is given by the piecewise function f.
> f:=piecewise(xL/2,2*h*(L-x)/L);
8
><
2 hx
x<
f := L
>:
L
2
2 h (L ¡ x) L
L 2 0.
> A[n]:=int(f*sin(n*Pi*x/L),x=0..L)/int(sin(n*Pi*x/L)^2,x=0..L)
assuming L>0:
The nth Fourier term in the in¯nite series representing the string displacement
is then Ãn =An Fn, the result being simpli¯ed by assuming that n is an integer.
> psi[n]:=simplify(A[n]*F[n]) assuming n::integer;
³
n¼
´ ³
n¼x
´ μ
cn¼t
8 h sin sin cos
2 L
L
Ãn :=
n2 ¼2 0